mlsanityData to 14 Oct 2026

A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs

arXiv 2001.04001 ↗First posted 12 Jan 2020math.NA cs.LG cs.NAcited by 105

Not on the map. Why?

Abstract (arXiv)

Traditional reduced order modeling techniques such as the reduced basis (RB) method (relying, e.g., on proper orthogonal decomposition (POD)) suffer from severe limitations when dealing with nonlinear time-dependent parametrized PDEs, because of the fundamental assumption of linear superimposition of modes they are based on. For this reason, in the case of problems featuring coherent structures that propagate over time such as transport, wave, or convection-dominated phenomena, the RB method usually yields inefficient reduced order models (ROMs) if one aims at obtaining reduced order approximations sufficiently accurate compared to the high-fidelity, full order model (FOM) solution. To overcome these limitations, in this work, we propose a new nonlinear approach to set reduced order models by exploiting deep learning (DL) algorithms. In the resulting nonlinear ROM, which we refer to as DL-ROM, both the nonlinear trial manifold (corresponding to the set of basis functions in a linear ROM) as well as the nonlinear reduced dynamics (corresponding to the projection stage in a linear ROM) are learned in a non-intrusive way by relying on DL algorithms; the latter are trained on a set of FOM solutions obtained for different parameter values. In this paper, we show how to construct a DL-ROM for both linear and nonlinear time-dependent parametrized PDEs; moreover, we assess its accuracy on test cases featuring different parametrized PDE problems. Numerical results indicate that DL-ROMs whose dimension is equal to the intrinsic dimensionality of the PDE solutions manifold are able to approximate the solution of parametrized PDEs in situations where a huge number of POD modes would be necessary to achieve the same degree of accuracy.

Key facts. Cited by 105 papers in the map (data through 14 Oct 2026). First posted on arXiv 12 Jan 2020.

Builds on

No reference list yet.

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 11; 2022: 14; 2023: 22; 2024: 19; 2025: 21; 2026: 18.

105 papers.

Sort:

By area:

  1. 2021POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decompositioncited by 79
  2. 2023A graph convolutional autoencoder approach to model order reduction for parametrized PDEscited by 44
  3. 2021Deep Learning for Reduced Order Modelling and Efficient Temporal Evolution of Fluid Simulationscited by 36
  4. 2022Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problemscited by 34
  5. 2021Multi-fidelity regression using artificial neural networks: efficient approximation of parameter-dependent output quantitiescited by 31
  6. 2022Reduced order modeling of parametrized systems through autoencoders and SINDy approach: continuation of periodic solutionscited by 29
  7. 2021A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equationscited by 28
  8. 2025Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraintscited by 28
  9. 2024Aero-Nef: Neural Fields for Rapid Aircraft Aerodynamics Simulationscited by 24
  10. 2025Reduced Order Modeling with Shallow Recurrent Decoder Networkscited by 24
  11. 2021Non-intrusive reduced order modeling of natural convection in porous media using convolutional autoencoders: comparison with linear subspace techniquescited by 23
  12. 2021A Comparison of Neural Network Architectures for Data-Driven Reduced-Order Modelingcited by 22
38 more
  1. 2021Accelerated replica exchange stochastic gradient Langevin diffusion enhanced Bayesian DeepONet for solving noisy parametric PDEscited by 22
  2. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  3. 2023Weak-Form Latent Space Dynamics Identificationcited by 20
  4. 2022Multi-fidelity surrogate modeling using long short-term memory networkscited by 18
  5. 2023Multi-fidelity reduced-order surrogate modelingcited by 18
  6. 2023Deep Learning-based surrogate models for parametrized PDEs: handling geometric variability through graph neural networkscited by 17
  7. 2023Learning Nonlinear Projections for Reduced-Order Modeling of Dynamical Systems using Constrained Autoencoderscited by 16
  8. 2023Deciphering and integrating invariants for neural operator learning with various physical mechanismscited by 15
  9. 2024On latent dynamics learning in nonlinear reduced order modelingcited by 14
  10. 2023Symplectic model reduction of Hamiltonian systems using data-driven quadratic manifoldscited by 13
  11. 2021Real-time simulation of parameter-dependent fluid flows through deep learning-based reduced order modelscited by 12
  12. 2022Reduced order modeling for flow and transport problems with Barlow Twins self-supervised learningcited by 12
  13. 2024Generalization Error Guaranteed Auto-Encoder-Based Nonlinear Model Reduction for Operator Learningcited by 12
  14. 2021An artificial neural network approach to bifurcating phenomena in computational fluid dynamicscited by 11
  15. 2022Operator inference with roll outs for learning reduced models from scarce and low-quality datacited by 11
  16. 2023Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representationcited by 11
  17. 2023Interpretable learning of effective dynamics for multiscale systemscited by 11
  18. 2022Long-time prediction of nonlinear parametrized dynamical systems by deep learning-based reduced order modelscited by 10
  19. 2024GFN: A graph feedforward network for resolution-invariant reduced operator learning in multifidelity applicationscited by 10
  20. 2023Latent assimilation with implicit neural representations for unknown dynamicscited by 9
  21. 2024A practical existence theorem for reduced order models based on convolutional autoencoderscited by 9
  22. 2022Predicting waves in fluids with deep neural networkcited by 7
  23. 2022Deep-HyROMnet: A deep learning-based operator approximation for hyper-reduction of nonlinear parametrized PDEscited by 7
  24. 2023Generative Adversarial Reduced Order Modellingcited by 7
  25. 2025Variationally correct operator learning: Reduced basis neural operator with a posteriori error estimationcited by 7
  26. 2023Low-dimensional Data-based Surrogate Model of a Continuum-mechanical Musculoskeletal System Based on Non-intrusive Model Order Reductioncited by 6
  27. 2023A Deep Learning algorithm to accelerate Algebraic Multigrid methods in Finite Element solvers of 3D elliptic PDEscited by 6
  28. 2023Learning the solution operator of two-dimensional incompressible Navier-Stokes equations using physics-aware convolutional neural networkscited by 6
  29. 2022textitFastSVD-ML-ROM: A Reduced-Order Modeling Framework based on Machine Learning for Real-Time Applicationscited by 5
  30. 2023Symplectic Autoencoders for Model Reduction of Hamiltonian Systemscited by 5
  31. 2021Deep neural network surrogates for non-smooth quantities of interest in shape uncertainty quantificationcited by 4
  32. 2021Data Assimilation Predictive GAN (DA-PredGAN): applied to determine the spread of COVID-19cited by 4
  33. 2023Latent Dynamics Networks (LDNets): learning the intrinsic dynamics of spatio-temporal processescited by 4
  34. 2023Active-Learning-Driven Surrogate Modeling for Efficient Simulation of Parametric Nonlinear Systemscited by 4
  35. 2024PTPI-DL-ROMs: pre-trained physics-informed deep learning-based reduced order models for nonlinear parametrized PDEscited by 4
  36. 2024Neural empirical interpolation method for nonlinear model reductioncited by 4
  37. 2024Real-time optimal control of high-dimensional parametrized systems by deep learning-based reduced order modelscited by 4
  38. 2025On the under-reaching phenomenon in message-passing neural PDE solvers: revisiting the CFL conditioncited by 4

50 most cited of 105 All 105 (JSON)